January 9, 2009

Jack and his wife went to a party where four other married couples were present. Every person shook hands with everyone he or she was not acquainted with. When the handshaking was over, Jack asked everyone, including his own wife, how many hands they shook. To his surprise, Jack got nine different answers. How many hands did Jack's wife shake ?

11 comments:

Secret Squïrrel
said...

It's easier if you draw it out with dots and lines.

Since no-one would have shaken hands with their partner, the most anyone could have shaken is 8. Nine diff answers means that the number of hands shaken range from 0 to 8 (and that Jack shook the same number as someone else).

I'll name each person by the number of hands that they shook. The partner of "8" must be the one who shook 0 hands since the other 8 people will each have shaken hands with "8". Similarly, the partner of "7" must be the one who shook hands once since everyone else (except "0") has now shaken hands at least twice (once with "8" and once with "7"). This same reasoning can be extended giving the following pairs:

8-07-16-25-34-4

Since we know that Jack shook the same number as someone else, he and his wife each shook 4 hands.

Oh, come on now! Jack and his wife went to a party where four other married couples were present. But how many single people were at the party? They shook hands with everyone they were not aquainted with. They were obviously invited to the party so they are probably aquainted with at least one other person. Even if somehow that was not true, we don't know how many single people were at the party.

Anonymous: You can figure out the number of hands Jack's wife shook without knowing how many hands he shook, as follows: Jack must have shaken the same number of hands as SOMEONE at the party is the only logical conclusion. When Jack asked the other nine partygoers how many hands each had shaken, he got nine different answers. There are only nine possible answers to the question, "How many hands did you shake?", the numbers from zero to eight. So since Jack had to give one of those answers, his answer had to be the same as one of the other nine guests. Since everyone except the "four-shaker" is already paired up with a spouse, Jack's wife must be the four-shaker. (Please note: It's also possible to demonstrate logically that Jack shook four hands, by a slightly more involved method of inference. Some versions of the puzzle ask the solver to determine how many hands Jack and his wife shook.)